I want to sew up one loose end here. Last time around I showed that this latest incarnation of the Rahmstorf model relating sea level to temperature was just as bogus at the previous versions. But I did not talk about one of their interesting (but ultimately irrelevant) new twists. Another layer of complexity was added by the application of Bayesian analysis, or in KMVR2011 nomenclature: “Bayesian multiple change-point regression.”

Bayesian analysis is a useful, but often counter intuitive, statistical method to tease out an underlying distribution from an observed distribution. That being said, the KMVR2011 application of Bayesian analysis starts out with a bogus model, which has been demonstrated ad nauseam. (See here and here.) This added layer of complexity simply obfuscates the failures of the starting model, rather that addressing those failures.

Please see this index of my posts concerning KMVR2011. Check back occasionally because the list of posts is slowly growing.

To(t), the “equilibrium temperature”

Recall the KMVR2011’s model includes a moving target “equilibrium temperure”, To(t), given by equation Ia
The “equilibrium temperature” can be determined by inserting the temperature history or scenario into equation Ia and solving the resulting differential equation for To(t). Figure 1, below, shows an equilibrium temperature found by KMVR2011 when Mann’s Global EIV land and ocean temperature is used.

Figure 1. this is figure 4C from KMVR2011

In my previous post I laid out a formula (equation II, previous post) for temperature vs. time that will cause the KMVR2011 model to yield an unrealistic sea level rise rate for a realistic temperature. In this post I will take the necessary step of finding the “equilibrium temperature” that results when my hypothetical temperature scenario is inserted into KMVR2011’s equation Ia. In a subsequent post I will show how my hypothetical temperature scenario and its resulting equilibrium temperature affect the sea level rise rate as calculated by the KMVR2011 model.

Quick and to the point

Here is To(t).

If you are not interested in the details, you can just take my word it and stop reading here. Otherwise, continue on the following sections.

“Reasonable” temperature scenarios

Even the best possible model could not be expected to give reasonable results if the input is nonsensical and it would not be a fair test of the model. That is why, for the moment, I am choosing to apply hypothetical temperatures for the past (1960 to 2000) to the KMVR2011 model. In that way the reader can compare my temperature scenarios to the same data used by KMVR2011 for that period and decide if my scenarios are “reasonable”.

The following graph shows five different temperature scenarios created by my temperature formula. Each of these scenarios is identical, except for the choice of γ (gamma).

Are these “reasonable” temperature scenarios? Are they a fair test of the KMVR2011 model? Let’s compare them to Hansen’s GISS instrumental temperature data and to Mann’s (Mann is the “M” in KMVR2011) own Global EIV, Land and Ocean temperature reconstruction for the same period…

To(t) from my hypothetical temperature scenarios

If you agree that my temperature scenarios are reasonable, then without further ado, here is the derivation of To(t).

Let

Inserting equation II into equation Ia gives

Letting

Then

Solving the differential equation in IIIa gives

The constant of integration, C2, can be found by choosing a known To(t) at some time, t’…

…and solving for C2

Now, simply substitute equation VI in equation IV for C2

Coming Soon

Sea level rise rates from the KMVR2011 model when my simple, reasonable temperature scenarios and the corresponding KMVR2011 “equilibrium temperatures” are used. I think you will find it interesting.

Update 11/27/11

The term (ατ + 1) were corrected to (ατ – 1) in equations (IV) through (VII). This was a typographical error and all calculations had been done with the correct term.

I would like to elaborate on my previous post, in which I presented a simple temperature vs. time function that causes KMVR2011’s model relating sea level rise rate to global temperature behave in a rather peculiar manner. I am trying to find a balance between simplicity, clarity and thoroughness. The level of mathematical literacy of my readers may vary widely, but this time around I need to employ some calculus. If the equations bother you, just consider the conclusions.

Starting with the conclusions

There exists a simple class of realistic temperature vs. time functions, which when applied to KMVR2011’s model yield results that disqualify it as representing a relationship between global temperature and sea level rise rate. This class of temperature vs. time functions gives a family of curves for which it is guaranteed that the higher the temperature the lower the sea level rise rate. This implausible effect is so severe that if forces rejection of the KMVR2011 model.

The Math

Here is the KMVR2011 model

where

Where H is the sea level, T(t) is the global temperature, Too, a1, a2, b and τ are all constants and To(t) is a to-be-determined time varying function related to T(t) as defined by equation Ia.

Now, consider the following temperature evolution. It is nearly the same as equation II from my previous post, but has an additional unitless constant, γ (a.k.a. “gamma”), in the exponential…

If equation II is inserted into equation I, then…

Rearranging terms in equation III gives…

H is the sea level. dH/dt, the derivative of the sea level, is the sea level rise rate. d2H/dt2, the second derivative of the sea level, is the rate at which the sea level rise rate changes. That is, d2H/dt2, is the acceleration. If d2H/dt2, is positive, the sea level rise rate is increasing. Conversely, if d2H/dt2, is negative, then the sea level rise rate is decreasing. Taking the time derivative of equation IIIa gives…

Let’s also consider the difference in the sea level rise rates at some time, t, for different values of γ. We can do this by analyzing the derivative of dH/dt (equation IIIa) with respect to γ.

What does the math tell us?

KMVR2011 does not conclude with specific values for their model constants and their time varying T0(t). Instead, they present probability density distributions for some constants, or combination of constants. However, there are some definite constraints that can be noted about the variables and their relationships to each other. These constraints are key to my conclusions.

Constraints:

a1 + a2 = a, where a1 and a2 are defined in KMVR2011 (see equation I, above) and a is defined in VR2009. VR2009 found a = 5.6 mm/yr/K.

a1 > 0 mm/yr/K and a2 > 0 mm/yr/K. KMVR20011 states that the distribution of a1 for their Bayesian analysis varied between 0.01 and 0.51 mm/yr/K. Needless to say, if either of these terms were less than zero the KMVR2011 model would make even less sense that it does now. That would be a road that the KMVR2011 authors do not want to travel.

As you can see from table 1, it gets little confusing for 0<γ<1. When a1, a2, b, C, and γ conform to the listed constraints, the signs of the various derivatives are known with certainty as long as…

But when …

at some point in time t- t’ will become large enough that d2H/dγdt will become positive. When that time occurs depends on the choices of a1, a2, b and γ. If we choose a1, a2 and b to agree with VR2009 (recall a1 +a2 = a = 5.6 mm/yr/K, and b = -49 mm/K) and γ = 0.8, then d2H/dγdt will continue to be negative until t – t’ = 44 years.

The conclusion, again.

Equation 2, above, can be used to build realistic hypothetical temperature evolutions. See figure 1, here, for some examples. Remember, KMR2011’s model relates sea level rise to temperature, and when applied to these hypothetical temperatures it must yield realistic sea level rises. It does not.

Table 1 shows various aspects of temperature and sea level using my hypothetical temperature evolution and KMVR2011’s resulting sea levels. Summed up succinctly, the table shows that with this combination the greater temperatures result in lower sea levels. This implausible situation disqualifies KMVR2011’s model.

Coming soon

I realize that a bunch of equations and a table do not give visceral understanding of this effect. A graphical illustration of these points will be coming soon.